Editing Lagrange point (section)

== Stability ==
[[File:Roche_potential.stl|thumb|upright=1.5|link=http://viewstl.com/classic/?embedded&url=https://upload.wikimedia.org/wikipedia/commons/8/86/Roche_potential.stl&shading=smooth&orientation=bottom&bgcolor=black|[[STL_(file_format)|STL 3D model]] of the Roche potential of two orbiting bodies, rendered half as a surface and half as a mesh]]
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Although the {{L1|nolink=yes}}, {{L2|nolink=yes}}, and {{L3|nolink=yes}} points are nominally unstable, there are quasi-stable periodic orbits called [[Halo orbit|''halo orbits'']] around these points in a three-body system. A full ''n''-body [[dynamical system]] such as the [[Solar System]] does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following [[Lissajous curve|Lissajous-curve]] trajectories. These quasi-periodic [[Lissajous orbit]]s are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of [[Orbital station-keeping|station keeping]] keeps a spacecraft in a desired Lissajous orbit for a long time.

For Sun–Earth-{{L1|nolink=yes}} missions, it is preferable for the spacecraft to be in a large-amplitude ({{convert|100000|–|200000|km|abbr=on|disp=or}}) Lissajous orbit around {{L1|nolink=yes}} than to stay at {{L1|nolink=yes}}, because the line between Sun and Earth has increased solar [[Interference (wave propagation)|interference]] on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around {{L2|nolink=yes}} keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.

The {{L4}} and {{L5}} points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25<ref group="note" name=exact_stability_threshold/> times the mass of the secondary body (e.g. the Moon),<ref name="Fitzpatrick">{{cite web|last1=Fitzpatrick|first1=Richard|title=Stability of Lagrange Points|url=http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node126.html|website=Newtonian Dynamics|publisher=University of Texas}}</ref><ref name="Greenspan">{{cite web|last1=Greenspan|first1=Thomas|title=Stability of the Lagrange Points, L4 and L5|url=http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf|date=7 January 2014|access-date=28 February 2018|archive-date=18 April 2018|archive-url=https://web.archive.org/web/20180418011225/http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf|url-status=dead}}</ref> The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth<ref name = "Pitjeva">{{cite journal |last1=Pitjeva |first1=Elena V. |author-link1=Elena V. Pitjeva |last2=Standish |first2=E. Myles |author-link2=E. Myles Standish |title=Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit |journal=Celestial Mechanics and Dynamical Astronomy |date=1 April 2009 |volume=103 |issue=4 |pages=365–372 |doi=10.1007/s10569-009-9203-8 |bibcode=2009CeMDA.103..365P |s2cid=121374703 |url=https://zenodo.org/record/1000691 }}</ref>). Although the {{L4|nolink=yes}} and {{L5|nolink=yes}} points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position, [[Coriolis acceleration]] (which depends on the velocity of an orbiting object and cannot be modeled as a contour map)<ref name="Greenspan" /> curves the trajectory into a path around (rather than away from) the point.<ref name="Greenspan" /><ref>Cacolici, Gianna Nicole, ''et al.,'' "[http://math.arizona.edu/~gabitov/teaching/141/math_485/Final_Report/Lagrange_Final_Report.pdf Stability of Lagrange Points: James Webb Space Telescope"], University of Arizona. Retrieved 17 Sept. 2018.</ref> Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around {{L4|nolink=yes}} and {{L5|nolink=yes}} are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.